Statistics: Posted by ExSan — Today, 12:35 pm

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Statistics: Posted by ExSan — Today, 12:35 pm

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[$]X_T \, = \, a_0 \, + a_1 \, Z_t \, + a_2 \, {Z_t}^2 \, + ...+ a_n \, {Z_t}^n[$]

[$]+\, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n[$]

but then I write the next equation after replacing [$]Z_t[$] series with [$]X_t[$] as

[$]X_T \,|X_t = \,X_t \, +\, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n[$]

which is wrong. The second equation implies that future variance is identical with respect to [$]X_t[$] which is totally counterintuitive since for example we know lognormal SDE diffusion in original coordinates takes very large diffusion in extreme right tail and very small diffusion in extreme left tail.

Point here is that future variance is identical with respect to [$]Z_t[$] coordinates but not with respect to [$]X_t[$] coordinates (that we see in reality as in lognormal SDE for example).

I believe when we switch from [$]Z_t [$] coordinates to [$]X_t[$] coordinates, the future variance has to be multiplied with [$]\frac{dX_t}{dZ_t}[$] as in

[$]X_T \,|X_t = \,X_t \, \, + \frac{dX_t}{dZ_t} \Big[ \, c_0 \, + \, c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n \Big][$]

This is a very quick and very tentative correction but I will come back with a complete corrected post in a day or two. Please read previous post in light of this tentative correction. And please pardon this mistake, I will fix it in a day.

Statistics: Posted by Amin — Today, 6:48 am

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How many votes does it take to kick Turkey out of NATO?

I’d like to bring this question up again. While they’re big and ugly and this may help scare off some common enemies, do we really need a pro-Russian muslim quasi-dictatorship in our friendly community? Especially since their main beef with the rest of us at the moment is that they hate free speech.Statistics: Posted by bearish — Yesterday, 10:39 pm

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- Variadics.pdf

Statistics: Posted by Cuchulainn — Yesterday, 8:44 pm

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Are they the Corrs?

Statistics: Posted by Cuchulainn — Yesterday, 8:34 pm

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First we suppose that we are doing monte carlo simulations of an asset according to its stochastic differential equation. This can possibly be a single lognormal/ CEV SDE or asset SDE in a correlated system with stochastic volatility.

Suppose we represent the asset SDE at any intermediate point t in monte carlo simulation by its Z-series representation as

[$]X_t \, = \, a_0 \, + a_1 \, Z_t \, + a_2 \, {Z_t}^2 \, + ...+ a_n \, {Z_t}^n[$]

This would have an equivalent hermite representation as

[$]X_t \, = \, ah_0 \, + ah_1 \,H_1( Z_t) \, + ah_2 \, H_2(Z_t) \, + ...+ ah_n \, H_n(Z_t)[$]

Now we suppose that asset SDE at terminal time related to option maturity is given by its Z-series as

[$]X_T \, = \, b_0 \, + b_1 \, Z_T \, + b_2 \, {Z_T}^2 \, + ...+ b_n \, {Z_T}^n[$]

which will have an equivalent hermite representation as

[$]X_T \, = \, bh_0 \, + bh_1 \,H_1( Z_T) \, + bh_2 \, H_2(Z_T) \, + ...+ bh_n \, H_n(Z_T)[$]

As I have been showing in previous post and also in my paper, we can find the stochastic evolution of SDE between intermediate time t and terminal time T as

[$]X_{t,T} \, = \, bh_0 \,- \, ah_0 + sgn(bh_1 - ah_1) \, \sqrt{sgn(bh_1) \, {bh_1}^2 - sgn(ah_1) \, {ah_1}^2} \,H_1( Z_{t,T}) \, + ...[$]

[$]+ sgn(bh_n - ah_n) \, \sqrt{sgn(bh_n )\, {bh_n}^2 - sgn(ah_n) \, {ah_n}^2} \,H_n( Z_{t,T}) \,[$]

Once we have calculated hermite representation of [$]X_{t,T}[$], we can also find its equivalent Z-series representation as

[$]X_{t,T} \, = \, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n[$]

Since [$]X_{T}[$] is sum of [$]X_{t}[$] and [$]X_{t,T}[$], we can write it as

[$]X_T \, = \, X_{t} \, + \, X_{t,T} [$]

which can be written as summation of respective Z-series as

[$]X_T \, = \, a_0 \, + a_1 \, Z_t \, + a_2 \, {Z_t}^2 \, + ...+ a_n \, {Z_t}^n[$]

[$]+\, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n[$]

To find the option prices and asset greeks at time [$]X_t[$] at time t, we realize that first line in above equation becomes conditional mean and second line becomes variance of the SDE at time t. So we can also write for a particular realized value of [$]X_t[$] in simulation as

[$]X_T \,|X_t = \,X_t \, +\, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n[$]

A call option with strike K can easily be found at time t given the particular value of [$]X_t[$] as

[$]\displaystyle\int_K^{\infty} \, (X_T \, - \, K) \, p(X_T) dX_T [$]

[$]=\displaystyle\int_{Z_K}^{\infty} \, (\,X_t \, +\, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n\, - \, K) \, p(Z_{t,T}) dZ_{t,T} [$]

where [$]Z_K[$] is different for every particular value of [$]X_t[$] and can be found from relation ship

[$]\, K \, - \, X_t \, = \, +\, c_0 \, + c_1 \, Z_{K} \, + c_2 \, {Z_{K}}^2 \, + ...+ c_n \, {Z_{K}}^n[$]

so [$] Z_K[$] is particular value of [$]Z_{t,T}[$] that equates [$]X_{t,T} \, = \, K \, - \, X_t \,[$]

Once [$]Z_K[$] has been found, we can convert the equation for call price given previously as

[$]=\displaystyle\int_{Z_K}^{\infty} \, (\,X_t \, +\, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n\, - \, K) \, p(Z_{t,T}) dZ_{t,T} [$]

into hermite polynomials and cheaply solve the equation quickly.

This way we can find option prices analytically along any simulation grid or path(More on how to do it in mote carlo setting later).

We can easily extend the technique to find delta and gamma greeks in middle of the simulation. Here is how.

We first write the equation for call option again as

[$]\displaystyle\int_K^{\infty} \, (X_T \, - \, K) \, p(X_T) dX_T [$]

[$]=\displaystyle\int_{Z_K}^{\infty} \, (X_T(Z_t,Z_{t,T}) \, - \, K) \, p(Z_{t,T}) dZ_{t,T} [$]

where in above equation [$]X_T(Z_t,Z_{t,T}) \,[$] is given as

[$]X_T(Z_t,Z_{t,T}) \, = \, a_0 \, + a_1 \, Z_t \, + a_2 \, {Z_t}^2 \, + ...+ a_n \, {Z_t}^n[$]

[$]+\, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n[$]

To find delta, we write

[$]\Delta =\frac{d \big[\displaystyle\int_{Z_K}^{\infty} \, (X_T(Z_t,Z_{t,T}) \, - \, K) \, p(Z_{t,T}) dZ_{t,T} \big]}{dX_t}[$]

[$]=\frac{d \big[\displaystyle\int_{Z_K}^{\infty} \, (X_T(Z_t,Z_{t,T}) \, - \, K) \, p(Z_{t,T}) dZ_{t,T} \big]}{dZ_t} \, \frac{dZ_t}{dX_t}[$]

So in order to calculate delta, we have to find the integral

[$]\frac{d \big[\displaystyle\int_{Z_K}^{\infty} \, (X_T(Z_t,Z_{t,T}) \, - \, K) \, p(Z_{t,T}) dZ_{t,T} \big]}{dZ_t} \, [$]

[$]=\displaystyle\int_{Z_K}^{\infty} \, \frac{d \big[X_T(Z_t,Z_{t,T}) \big]}{dZ_t} \, p(Z_{t,T}) dZ_{t,T} \,- \frac{dZ_K}{dZ_t} (X_T(Z_t,Z_K) \, - \, K) \, p(Z_{K}) [$]

For a vanilla call option, the second term of Liebniz integral rule goes to zero since [$](X_T(Z_t,Z_K) \, - \, K)[$] goes to zero because [$]X_T(Z_t,Z_K) \, = \, K[$]

and we are left with solving only the integral below as

[$]\displaystyle\int_{Z_K}^{\infty} \, \frac{d \big[X_T(Z_t,Z_{t,T})]}{dZ_t} \, p(Z_{t,T}) dZ_{t,T} [$]

since

[$]X_T(Z_t,Z_{t,T}) \, = \, a_0 \, + a_1 \, Z_t \, + a_2 \, {Z_t}^2 \, + ...+ a_n \, {Z_t}^n[$]

[$]+\, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n[$]

We have

[$]\displaystyle\int_{Z_K}^{\infty} \, \frac{d \big[ (\, a_0 \, + a_1 \, Z_t \, + a_2 \, {Z_t}^2 \, + ...+ a_n \, {Z_t}^n)+\, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n \big]}{dZ_t} \, p(Z_{t,T}) dZ_{t,T} [$]

differentiating with respect to [$]Z_t[$] is relevant for the conditional mean only and performing the differentiation, our new equation becomes

[$]\displaystyle\int_{Z_K}^{\infty} \, \big[ \,(a_1 \, +2 a_2 \, Z_t \, + ...+n \, a_n \, {Z_t}^{n-1})+\, c_0 \, + c_1 \, Z_{t,T} \, + c_2 \, {Z_{t,T}}^2 \, + ...+ c_n \, {Z_{t,T}}^n \, \big] \, p(Z_{t,T}) dZ_{t,T} [$]

Above integral can be easily integrated analytically. Again the boundary value of [$]Z_K[$] is found differently for every particular value of [$]X_t[$] from relationship

[$]\, K \, - \, X_t \, = \, +\, c_0 \, + c_1 \, Z_{K} \, + c_2 \, {Z_{K}}^2 \, + ...+ c_n \, {Z_{K}}^n[$]

so [$] Z_K[$] is particular value of [$]Z_{t,T}[$] that equates [$]X_{t,T} \, = \, K \, - \, X_t \,[$]

Calculating the last integral would find the value for delta for call option at a particular point [$]X_t[$] on a simulation grid after substituting it in previous relevant equation.

We can find gamma by repeating the procedure for delta but we might have to include the Liebniz integral rule term in gamma calculations that went to zero for calculations of delta.

An important point for monte carlo simulations is that we can probably do two simulations. In first simulation, we can calculate the Z_series parameters on every time step and also calculate all variances [$]X{t,T}[$]. For monte carlo simulations, it would be convenient to form a uniform grid(or variable grid if you can) and find option prices and deltas on the grid using above procedure. In later simulation, option prices and delta, gammas can be found by interpolation of monte carlo simulated values between the grid points. I did something like this in a different setting in one of recent programs that I posted on Wilmott.

I will try to write a program about it and post it on Wilmott in next few days.

I also have an antipsychotic injection due tomorrow and I may not remain lucid for another 7-10 days after that. I want to request friends again to please protest to American government and defense for forcing innocent people who just want to do science on antipsychotic injections.

Statistics: Posted by Amin — Yesterday, 8:17 pm

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Statistics: Posted by tagoma — Yesterday, 5:35 pm

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Maybe just measure how frequently Amazon drivers leave your valuable deliveries on the doorstep of a busy London street.

Using that as a Stupid Index assumes they care whether your parcels are stolen or not.Apparently UPS (do they have that in Britain? It's a delivery service, probably the oldest ongoing and the biggest in America) originally required signatures for receipt of deliveries, which meant that if no one was available to sign, they'd leave a note and try to deliver another day. And then after a few failed attempts to deliver, the note would say where you had to go to collect your package.

But they decided -- hopefully based on actual data -- that it was more efficient, even with thefts, to just leave deliveries at the door when no one was there to receive them.

Now you can request deliveries with signature required, and they require signatures to receive booze, but the standard is they just leave it.

Similarly, it can be surmised that credit card fraud is not a serious problem, because credit card companies -- and at least in America, you are generally limited to a $50 loss in the event of credit card fraud, and I think usually they won't hit you even with that -- are very slow to bother to do anything about it: in America, the chip-in-card system that was broadly used in Europe was not adopted for at least ten years after the technology was available and had been thoroughly tested.

Statistics: Posted by Marsden — Yesterday, 5:31 pm

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Well, except Delphine Claudel. I’d be OK being lumped in with her.

She has been performing great lately, indeed!Les Rousses is only 25km away from here.

Statistics: Posted by tagoma — Yesterday, 9:14 am

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Maybe just measure how frequently Amazon drivers leave your valuable deliveries on the doorstep of a busy London street.

I suspect it would be highly correlated with people born since the invention of the internet.

Statistics: Posted by Paul — Yesterday, 7:49 am

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1. hedged simulations where we would have access to various greeks of particular derivative product in parametric form within simulations and we could find our P/L and other risk along the evolution of any simulated monte carlo path.

2. Modelling of volatility surfaces in parametric forms.

3. Modelling of risk-neutral densities especially the challenging bi-modal densities in parametric form.

4. Trading and hedging of variance swaps by making a statistical comparison of implied and realized volatility.

5. Trading of options in general by making a statistical comparison of various factors affecting the option prices as between factors realized from risk-neutral data and the same factors realized from the underlying markets and gaining exposure to the particular factor that is misaligned between the two markets while hedging other factors that affect the option pricing.

6. Developing trading strategies that employ a combination of trading in a specific type of exotic option while simultaneously trading the underlying or other simple options.

I will be sharing all the codes and insights with friends on this forum as usual.

I want to work on application of our newly developed tools towards above and so many other possibilities in hedging and trading of options and other derivatives for next 4-5 months. After that I want to start shifting towards AI so that we will start using our insights and understanding in further new ways. I really regret that I stopped working on my research for 2-3 months close to the end of the last year and wish that I had continued to work on stochastics research even when I was working with algorithmic trading.

Statistics: Posted by Amin — Yesterday, 6:41 am

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